The $h-$th power of every element in a finite group of order $h$ is the identity element of the group

Question

I work in population dynamics. I want to show somewhere in my work that the $h-$th power of every element in a finite group of order $h$ is the identity element of the group. I guess this is elementary result but it would be nice if somebody show me how to do it or where to find it. Thanks

Answer 1

The order of every element of a group is a divisor of $h$ by Lagrange's Theorem. So if $x$ is an element of your group of order $k$ then $h = k\cdot l$ for some $l.$ Hence we have

$$x^h = x^{k\cdot l} = (x^k)^l = (e)^l = e.$$